let i be Element of NAT ; :: thesis: for G being Go-board st i <= len G holds
not cell G,i,(width G) is Bounded
let G be Go-board; :: thesis: ( i <= len G implies not cell G,i,(width G) is Bounded )
assume A1: i <= len G ; :: thesis: not cell G,i,(width G) is Bounded
per cases ( i = 0 or ( i >= 1 & i < len G ) or i = len G ) by A1, NAT_1:14, XXREAL_0:1;
suppose A2: i = 0 ; :: thesis: not cell G,i,(width G) is Bounded
A3: cell G,0 ,(width G) = { |[r,s]| where r, s is Element of REAL : ( r <= (G * 1,1) `1 & (G * 1,(width G)) `2 <= s ) } by GOBRD11:25;
for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell G,0 ,(width G) & not |.q.| < r )
proof
let r be Real; :: thesis: ex q being Point of (TOP-REAL 2) st
( q in cell G,0 ,(width G) & not |.q.| < r )
take q = |[(min (- r),((G * 1,1) `1 )),((G * 1,(width G)) `2 )]|; :: thesis: ( q in cell G,0 ,(width G) & not |.q.| < r )
A4: abs (q `1 ) <= |.q.| by JGRAPH_1:50;
min (- r),((G * 1,1) `1 ) <= (G * 1,1) `1 by XXREAL_0:17;
hence q in cell G,0 ,(width G) by A3; :: thesis: not |.q.| < r
per cases ( r <= 0 or r > 0 ) ;
suppose A5: r > 0 ; :: thesis: not |.q.| < r
q `1 = min (- r),((G * 1,1) `1 ) by EUCLID:56;
then A6: abs (- r) <= abs (q `1 ) by A5, TOPREAL6:8, XXREAL_0:17;
- (- r) > 0 by A5;
then - r < 0 ;
then - (- r) <= abs (q `1 ) by A6, ABSVALUE:def 1;
hence not |.q.| < r by A4, XXREAL_0:2; :: thesis: verum
end;
end;
end;
hence not cell G,i,(width G) is Bounded by A2, JORDAN2C:40; :: thesis: verum
end;
suppose A7: ( i >= 1 & i < len G ) ; :: thesis: not cell G,i,(width G) is Bounded
then A8: cell G,i,(width G) = { |[r,s]| where r, s is Element of REAL : ( (G * i,1) `1 <= r & r <= (G * (i + 1),1) `1 & (G * 1,(width G)) `2 <= s ) } by GOBRD11:31;
for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell G,i,(width G) & not |.q.| < r )
proof
let r be Real; :: thesis: ex q being Point of (TOP-REAL 2) st
( q in cell G,i,(width G) & not |.q.| < r )
take q = |[((G * i,1) `1 ),(max r,((G * 1,(width G)) `2 ))]|; :: thesis: ( q in cell G,i,(width G) & not |.q.| < r )
A9: max r,((G * 1,(width G)) `2 ) >= (G * 1,(width G)) `2 by XXREAL_0:25;
width G <> 0 by GOBOARD1:def 5;
then A10: 1 <= width G by NAT_1:14;
( i < i + 1 & i + 1 <= len G ) by A7, NAT_1:13;
then (G * i,1) `1 <= (G * (i + 1),1) `1 by A7, A10, GOBOARD5:4;
hence q in cell G,i,(width G) by A8, A9; :: thesis: not |.q.| < r
A11: abs (q `2 ) <= |.q.| by JGRAPH_1:50;
end;
hence not cell G,i,(width G) is Bounded by JORDAN2C:40; :: thesis: verum
end;
suppose A13: i = len G ; :: thesis: not cell G,i,(width G) is Bounded
A14: cell G,(len G),(width G) = { |[r,s]| where r, s is Real : ( (G * (len G),1) `1 <= r & (G * 1,(width G)) `2 <= s ) } by GOBRD11:28;
for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell G,i,(width G) & not |.q.| < r )
proof
let r be Real; :: thesis: ex q being Point of (TOP-REAL 2) st
( q in cell G,i,(width G) & not |.q.| < r )
take q = |[(max r,((G * (len G),1) `1 )),((G * 1,(width G)) `2 )]|; :: thesis: ( q in cell G,i,(width G) & not |.q.| < r )
A15: abs (q `1 ) <= |.q.| by JGRAPH_1:50;
(G * (len G),1) `1 <= max r,((G * (len G),1) `1 ) by XXREAL_0:25;
hence q in cell G,i,(width G) by A13, A14; :: thesis: not |.q.| < r
end;
hence not cell G,i,(width G) is Bounded by JORDAN2C:40; :: thesis: verum
end;
end;